Main differences between small and large molecule (mAb) PBPK models
- mAb clearance is controlled by interaction with endosomal FcRn
- mAbs typically don’t enter the intracellular space
- Lymphatic system is crucial for mAb drainage to circulation
A bottom-up modeling approach that uses physiological parameters to describe a drug’s pharmacokinetics. The concept of PBPK modeling was introduced by Torsten Teorell in 1937 https://www.tandfonline.com/doi/pdf/10.3109/03009739509178895
https://www.mdpi.com/1999-4923/9/4/41/htm
# load libraries
library(tidyverse)
library(mrgsolve)
\[\frac{dA_{mu}}{dt}=Q_{mu}(C_{art}-\frac{C_{mu}}{\frac{Kp_{mu}}{BP}})\]
\[\frac{dA_{li}}{dt}=Q_{li}(C_{art}-\frac{C_{li}}{\frac{Kp_{li}}{BP}})-Cl_{li}.f_{u}.\frac{C_{li}}{\frac{Kp_{li}}{BP}}\]
\[\frac{dA_{art}}{dt}=Q_{lu}(\frac{C_{lu}}{\frac{Kp_{lu}}{BP}}-C_{art})\]
\[\frac{dA_{ven}}{dt}=\sum_{T\neq lu} (Q_T.\frac{C_T}{\frac{Kp_T}{BP}}) − Q_{lu}.C_{ven}\]
\[\frac{dA_{lu}}{dt}=Q_{lu}(C_{ven} − \frac{C_{lu}}{\frac{Kp_{lu}}{BP}})\]
Use models/simplePBPK.mod file to build a simple
PBPK model.
Use script.R script to compile the model and run a
simple simulation (chunks 1 and 2).
Zane NR, Thakker DR. A physiologically based pharmacokinetic model
for voriconazole disposition predicts intestinal first-pass metabolism
in children. Clin Pharmacokinet. 2014;53: 1171–1182
https://link.springer.com/article/10.1007%2Fs40262-014-0181-y
ICRP Publication 89 http://www.icrp.org/publication.asp?id=ICRP%20Publication%2089
Utsey, Kiersten, Madeleine S. Gastonguay, Sean Russell, Reed Freling, Matthew M. Riggs, and Ahmed Elmokadem. 2020. “Quantification of the Impact of Partition Coefficient Prediction Methods on PBPK Model Output Using a Standardized Tissue Composition.” Drug Metabolism and Disposition: The Biological Fate of Chemicals, July. https://doi.org/10.1124/dmd.120.090498.
Note: for the following tasks use
script.R
models/voriPBPK.modmodA <- mread("models/voriPBPK.mod")
calcKp_PT.R function to calculate voriconzole
tissue:plasma partition coefficients according to Poulin and Theil
method https://www.ncbi.nlm.nih.gov/pubmed/11782904.source("calcKp_PT.R")
#voriconazole physicochemical properties
logP <- 2.56 #lipophilicity
pKa <- 1.76
fup <- 0.42 #unbound fraction in plasma
type <- 3 #monoprotic base
BP <- 1 #blood:plasma concentration ratio
dat <- read.csv("data/tissue_comp_PT.csv")
#calculate partition coefficients
Kp <- calcKp_PT(logP=logP, pKa=pKa, fup=fup, BP=BP, type=type, dat=dat)
#update model parameters partition coefficients
modA <- param(modA, Kp)
Adult_IV.csv. (N.B.: observed data were digitized from Zane
and Thakker (2014) paper using WebPlotDigitizer https://automeris.io/WebPlotDigitizer/):#load observed IV infusion data
obs <- read.csv("data/Adult_IV.csv")
#set simulation conditions
bw <- 73
amt <- 4*bw
rate <- 4*bw
cmt <- "VEN"
ii <- 12
addl <- 13
ss <- 1
#run simulation
sim <-
modA %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss) %>%
mrgsim(delta = 0.1, end = 12) %>%
filter(row_number() != 1)
#plot prediction and compare to observed data
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) +
scale_colour_manual(name='',
values=c('sim'='black', 'observed'='black'),
breaks=c("observed","sim"),
labels=c("observed","predicted")) +
guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
labs(title="Adult 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
Adult_PO.csv.obs <- read.csv("data/Adult_PO.csv")
bw <- 73
amt <- 200
cmt <- "GUTLUMEN"
ii <- 12
addl <- 13
ss <- 1
sim <-
modA %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=ss) %>%
mrgsim(delta = 0.1, end = 12) %>%
filter(row_number() != 1)
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) +
scale_colour_manual(name='',
values=c('sim'='black', 'observed'='black'),
breaks=c("observed","sim"),
labels=c("observed","predicted")) +
guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
labs(title="Adult 200 mg PO", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
modA <- mread("models/voriPBPK_ext.mod") %>%
param(MPPGI = 30.3/30) %>%
param(Kp)
sim <-
modA %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=ss) %>%
mrgsim(delta = 0.1, end = 12) %>%
filter(row_number() != 1)
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) +
scale_colour_manual(name='',
values=c('sim'='black', 'observed'='black'),
breaks=c("observed","sim"),
labels=c("observed","predicted")) +
guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
labs(title="Adult 200 mg PO", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
# pediatric (5 yo) male physiology; https://www.ncbi.nlm.nih.gov/pubmed/14506981
pedPhys <- list(WEIGHT = 19,
Vad = 5.5,
Vbo = 2.43,
Vbr = 1.31,
VguWall = 0.22,
VguLumen = 0.117,
Vhe = 0.085,
Vki = 0.11,
Vli = 0.467,
Vlu = 0.125,
Vmu = 5.6,
Vsp = 0.05,
Vbl = 1.5,
Qad = 0.05*3.4*60,
Qbo = 0.05*3.4*60,
Qbr = 0.12*3.4*60,
Qgu = 0.15*3.4*60,
Qhe = 0.04*3.4*60,
Qki = 0.19*3.4*60,
Qmu = 0.17*3.4*60,
Qsp = 0.03*3.4*60,
Qha = 0.065*3.4*60,
Qlu = 3.4*60,
MPPGL = 26,
VmaxH = 120.5,
KmH = 11,
MPPGI = 0,
VmaxG = 120.5,
KmG = 11)
modP <- param(modA, pedPhys)
Pediatric_IV.csv.obs <- read.csv("data/Pediatric_IV.csv") #load observed data
wt <- 19 #pediatric body weight
amt <- 4*wt
rate <- 3*wt
cmt <- "VEN" #intravenous infusion
ii <- 12
addl <- 13
ss <- 1
# simulate
sim <-
modP %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>%
mrgsim(delta = 0.1, end = 12) %>%
dplyr::filter(row_number() != 1)
# plot
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) +
scale_colour_manual(name='',
values=c('sim'='black', 'observed'='black'),
breaks=c("observed","sim"),
labels=c("observed","predicted")) +
guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
labs(title="Pediatric 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
Pediatric_PO.csv Note: Include a similar 30-fold
lower intestinal clearance than hepatic clearance.obs <- read.csv("data/Pediatric_PO.csv") #load observed data
# adjust intestinal clearance
modP <- modP %>% param(MPPGI = 26 / 30)
# simulation conditions
bw <- 19
amt <- 4 * bw
cmt <- "GUTLUMEN"
ii <- 12
addl <- 13
ss <- 1
# simulate
sim <-
modP %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=1) %>%
mrgsim(delta = 0.1, end = 12) %>%
dplyr::filter(row_number() != 1)
# plot
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) +
scale_colour_manual(name='',
values=c('sim'='black', 'observed'='black'),
breaks=c("observed","sim"),
labels=c("observed","predicted")) +
guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
labs(title="Pediatric 4 mg/kg PO", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
Run graphical sensitivity analysis for the muscle:plasma
(Kpmu) and lung:plasma (Kplu) partition
coefficients using adult IV data.
#set simulation conditions
bw <- 73
amt <- 4*bw
rate <- 4*bw
cmt <- "VEN"
ii <- 12
addl <- 13
ss <- 1
##' Define an intervention
e <- ev(cmt=cmt, amt=amt, rate=rate, ii= ii, addl=addl, ss=1)
## Sensitivity analysis on Kpmu
idata <- expand.idata(Kpmu = c(3/2, 3, 3*2))
modA %>%
carry_out(Kpmu) %>%
mrgsim_ei(e, idata, delta = 0.1, recsort=3, obsonly=TRUE, end = 12) %>%
mutate(Kpmu = factor(Kpmu)) %>%
ggplot(aes(x=time, y=CP, col=Kpmu)) +
geom_line() +
labs(x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
##' Sensitivity analysis on Kplu
idata <- expand.idata(Kplu = c(1/2, 1, 1*2))
modA %>%
carry_out(Kplu) %>%
mrgsim_ei(e, idata, delta = 0.1, recsort=3, obsonly=TRUE, end = 12) %>%
mutate(Kplu = factor(Kplu)) %>%
ggplot(aes(x=time, y=CP, col=Kplu)) +
geom_line() +
labs(x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
mrgsim.sa packageThe package mrgsim.sa simplifies the process of running local sensitivity analysis https://github.com/kylebaron/mrgsim.sa. The package can do graphical sensitivity as well as obtaining local sensitivity coefficients.
\[\frac{\partial{y_i}}{\partial{\theta_j}}.\frac{w_{\theta{j}}}{w_{y_i}}\]
What is the most influential parameter?
library(mrgsim.sa)
### sensitivity analysis
#set the output variable of interest
sensvar <- c("CP")
# graphical sensitivity for each parameter
out_sens <-
modA %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss) %>%
select_par(all_of(names(Kp[-11]))) %>%
parseq_fct(.n=3) %>%
sens_each(delta = 0.1, recsort=3, obsonly=TRUE, end = 12)
sens_plot(out_sens, "CP")
# graphical sensitivity for a grid of parameters
out_sens_grid <-
modA %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss) %>%
parseq_cv(Kpmu, Kplu, .n=3) %>%
sens_grid(delta = 0.1, recsort=3, obsonly=TRUE, end = 12)
sens_plot(out_sens_grid, "CP")
# local sensitivity analysis
out_lsa <- lsa(modA, var = "CP", par = names(all_of(Kp[-11])), events = ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss), end=12, delta=0.1, eps=1e-4)
lsa_plot(out_lsa, pal = NULL)
# get a summary of sensitivity coefficients
out_lsa_summ <- out_lsa %>%
group_by(p_name) %>%
summarise(mean_sens = mean(sens)) %>%
ungroup()
ggplot(data=out_lsa_summ, aes(x=reorder(p_name, mean_sens), y=mean_sens)) +
geom_col() +
labs(x="Parameter", y="Coefficient") +
coord_flip() +
geom_hline(yintercept = 0, lty=2) +
theme_bw()
Use the package sensobol (https://cran.r-project.org/web/packages/sensobol/index.html)
to run a global sensitivity analysis using the Sobol method.
sensobol reference: Puy, Arnald, Samuele Lo Piano, Andrea Saltelli,
and Simon A. Levin. 2021. “Sensobol: An R Package to Compute
Variance-Based Sensitivity Indices.” arXiv [stat.CO]. arXiv. http://arxiv.org/abs/2101.10103.
General sobol sensitivity method: Zhang, X-Y, M. N. Trame, L. J.
Lesko, and S. Schmidt. 2015. “Sobol Sensitivity Analysis: A Tool to
Guide the Development and Evaluation of Systems Pharmacology Models.”
CPT: Pharmacometrics & Systems Pharmacology 4 (2): 69–79. https://ascpt.onlinelibrary.wiley.com/doi/full/10.1002/psp4.6
Vignette: https://github.com/mrgsolve/gallery/blob/master/application/global-sensobol.md
# library(sensobol)
# library(future)
# library(mrgsim.parallel)
#
# # set parallelization options
# nCores <- future::availableCores()
# options(future.fork.enable=TRUE, mc.cores = nCores)
# plan(multicore, workers = nCores)
#
# # generate parameter sets
# N <- 1e5 # initial sample number
# mat <- sobol_matrices(N = N, params = c("Kpmu","Kplu","Kpli","Kpsp")) # creates sample variates between 0-1
# head(mat)
#
# #Transform and groom
# params <- unlist(as.list(param(modA))[c("Kpmu","Kplu","Kpli","Kpsp")])
# umin <- params / 3
# umax <- params * 3
# umin
# umax
#
# # get samples by applying quantile to the uniform distributed matrix
# mat <- as_tibble(mat)
# mat2 <- imodify(mat, ~ qunif(.x, umin[[.y]], umax[[.y]]))
# mat2 <- mutate(mat2, ID = row_number())
# head(mat2)
#
# # run the analysis
# ## serial
# system.time(out <- modA %>% future_mrgsim_ei(e, mat2, nchunk = nCores, end = 1, delta = 1, Request = c("CP"), rtol = 1e-4, output = "df"))
#
# # calculate Cmax
# y <- out %>% group_by(ID) %>% summarise(cmax = max(CP)) %>% ungroup()
# y <- as.numeric(y$cmax)
#
# # get indices
# ind <- sobol_indices(Y = y, N = N, params = names(params), boot = TRUE, R = 1000, first = "jansen")
# ind.dummy <- sobol_dummy(Y = y, N = N, params = names(params), boot = TRUE, R = 1000) # dummy to assess numerical approximation error
#
# # plot indices
# plot(ind, dummy = ind.dummy) + ylim(0,1)
nloptr (https://cran.r-project.org/web/packages/nloptr/index.html)
to optimize for most influential partition coefficient parameter and
compare prediction before and after optimization to observed data
numDeriv (https://cran.r-project.org/web/packages/numDeriv/index.html)
to generate the 95% CI around the parameter estimates library(nloptr)
library(numDeriv)
library(kableExtra)
### Do some data assembly to create dataset for optimization
#load observed IV infusion data
bw <- 73
obs <- read.csv("data/Adult_IV.csv")
# create an nm-tran dataset
nm_dv <- obs %>%
select(time, dv=obs) %>%
mutate(ID = 1,
amt = 0,
rate = 0,
cmt = "VEN",
evid = 0,
ii = 0,
addl = 0,
ss = 0)
nm_dose <- nm_dv %>%
slice(1) %>%
mutate(amt = 4*bw,
rate = 4*bw,
ii = 12,
addl = 13,
ss = 1,
dv = NA,
evid = 1)
nm <- bind_rows(nm_dose, nm_dv) %>% arrange(time)
##set up objective function
OF <- function(pars, dat, pred=F){
pars <- lapply(pars,exp) #Get out of log domain for MLE
pars <- as.list(pars)
names(pars) <- names(theta)
## Get a prediction
out <- modA %>% param(pars) %>% mrgsim_d(dat, carry_out=c("dv"), output="df")
if(pred) return(out)
##OLS
#return(sum((out$CP - out$dv)^2))
##maximum likelihood
return(-1*sum(dnorm(log(out$dv),
mean=log(out$CP),
sd=pars$sigma, log=TRUE), na.rm = T))
}
# set initial values
theta <- log(c(Kpmu=2.94, sigma=1)) #initial parameter for MLE; mean and standard deviation
##Fit with nloptr package
##derivative-free optimizers
#fit <- neldermead(theta, OF, dat=nm) #Nelder-Mead simplex
fit <- newuoa(theta, OF, dat=nm) #New Unconstrained Optimization with quadratic Approximation
##gradient-basd optimizers
#fit <- tnewton(theta, OF, dat=nm) #Local optimizer; Nelder-Mead simplex
#fit <- mlsl(theta, OF, dat=nm, lower=log(c(0.1, 0.1)), upper=log(c(10, 2))) #global optimizer; multi-level single-linkage; takes very long ~ 10 min
##global optimizers
#fit <- direct(OF, dat=nm, lower=log(c(0.1, 0.1)), upper=log(c(10, 2))) #DIviding RECTangles algorithm; takes ~ 3 min
fit
## $par
## [1] -0.5708297 -2.0483168
##
## $value
## [1] -8.181457
##
## $iter
## [1] 51
##
## $convergence
## [1] 4
##
## $message
## [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
p <- as.list(exp(fit$par)) #get the parameters on the linear scale
names(p) <- names(theta)
p
## $Kpmu
## [1] 0.5650564
##
## $sigma
## [1] 0.1289518
# get standard error and confidence intervals around the estimated parameters
h <- hessian(OF, fit$par, dat=nm)
vc_log <- solve(h) #variance-covariance matrix
SE_log <- sqrt(diag(vc_log)) #standard error on log scale
# create dataframe with parameter summary
sig <- function(x) signif(x, 3)
paramSumm <- tibble(Parameter = names(theta),
Estimate = exp(fit$par),
lb = exp(fit$par - (1.96 * SE_log)),
ub = exp(fit$par + (1.96 * SE_log))) %>%
mutate(`Estimate (95% CI)` = paste0(sig(Estimate), " (", sig(lb), ", ", sig(ub), ")")) %>%
select(Parameter, `Estimate (95% CI)`)
paramSumm %>%
knitr::kable() %>%
kable_styling()
| Parameter | Estimate (95% CI) |
|---|---|
| Kpmu | 0.565 (0.202, 1.58) |
| sigma | 0.129 (0.0878, 0.189) |
# compare initial predictions to those using the optimized parameters
predB4 <- OF(theta, dat=nm, pred=T)
predAfter <- OF(fit$par, dat=nm, pred=T)
gp <- ggplot() +
geom_point(data=obs, aes(x=time, y=obs)) +
geom_errorbar(data=obs, aes(x=time, y=obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data=predB4, aes(x=time, y=CP), lty=2) +
geom_line(data=predAfter, aes(x=time, y=CP)) +
labs(x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
check out:
nlmixr package for nonlinear mixed effects modeling https://github.com/nlmixrdevelopment/nlmixrSimulate a 4 mg/kg voriconazole IV infusion dosing in a male child
subject infused at a rate of 3 mg/kg/h twice a day for seven days.
Compare the steady state prediction before and after optimization to the
observed data in Pediatric_IV.csv.
obs <- read.csv("data/Pediatric_IV.csv") #load observed data
wt <- 19 #adult body weight
amt <- 4*wt
rate <- 3*wt
cmt <- "VEN" #intravenous infusion
ii = 12
addl = 13
ss = 1
# simulate
simB4 <- as.data.frame(modP %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>%
mrgsim(delta = 0.1, end = 12)) %>%
dplyr::filter(row_number() != 1)
simAfter <- as.data.frame(modP %>%
param(Kpmu=p$Kpmu) %>%
ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>%
mrgsim(delta = 0.1, end = 12)) %>%
dplyr::filter(row_number() != 1)
# plot
gp <- ggplot() +
geom_point(data = obs, aes(x=time, y=obs), size=2.5) +
geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
geom_line(data = simB4, aes(x=time, y=CP), lty=2) +
geom_line(data = simAfter, aes(x=time, y=CP)) +
labs(title="Pediatric 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
Need to introduce interindividual variability to physiological parameters
Willmann, Stefan, Karsten Höhn, Andrea Edginton, Michael Sevestre, Juri Solodenko, Wolfgang Weiss, Jörg Lippert, and Walter Schmitt. 2007. “Development of a Physiology-Based Whole-Body Population Model for Assessing the Influence of Individual Variability on the Pharmacokinetics of Drugs.” Journal of Pharmacokinetics and Pharmacodynamics 34 (3): 401–31. https://pubmed.ncbi.nlm.nih.gov/17431751/
Huisinga, W., A. Solms, L. Fronton, and S. Pilari. 2012. “Modeling Interindividual Variability in Physiologically Based Pharmacokinetics and Its Link to Mechanistic Covariate Modeling.” CPT: Pharmacometrics & Systems Pharmacology 1 (September): e4. https://ascpt.onlinelibrary.wiley.com/doi/10.1038/psp.2012.3
“NHANES - National Health and Nutrition Examination Survey Homepage.” 2018. July 26, 2018. http://www.cdc.gov/nchs/nhanes/.
Meyer, Michaela, Sebastian Schneckener, Bernd Ludewig, Lars Kuepfer, and Joerg Lippert. 2012. “Using Expression Data for Quantification of Active Processes in Physiologically Based Pharmacokinetic Modeling.” Drug Metabolism and Disposition: The Biological Fate of Chemicals 40 (5): 892–901. https://pubmed.ncbi.nlm.nih.gov/22293118/
../data/derived/popPars_100.rds# load population params
popPars <- readRDS("data/popPars_100.rds")
# add IIV on CL/VmaxH
set.seed(192898)
iVmaxH <- rlnorm(100, meanlog=log(40), sdlog=0.2)
# add to popPars
popPars2 <- lapply(1:length(popPars), function(i){
pars <- c(popPars[[i]],list(VmaxH = iVmaxH[i]))
return(pars)
})
# simulate
modA2 <- mread("models/voriPBPK2.mod")
Kpmu <- 0.56
modA2 <- param(modA2, Kpmu=Kpmu)
#set simulation conditions
bw <- 73
amt <- 4*bw
rate <- 4*bw
cmt <- "VEN"
ii <- 12
addl <- 13
ss <- 1
delta <- 0.1
end <- 12
# prepare simulation dataset
idata <- popPars2 %>% bind_rows()
e <- ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss)
data <- e %>%
as_tibble %>%
bind_cols(idata) %>%
mutate(amt = 4*BW, rate = 4*BW) %>%
select(ID, everything())
#run simulation
system.time(sims <- modA2 %>%
mrgsim_d(data=data, delta = delta, end = end, obsonly=T, outvars = c("CP"), output="df"))
## user system elapsed
## 2.676 0.003 2.678
# get summary stats for population prediction
hi95 <- function(x) quantile(x, probs = c(0.95))
lo05 <- function(x) quantile(x, probs = c(0.05))
sims2 <- sims %>%
group_by(time) %>%
mutate(loCP = lo05(CP),
medCP = median(CP),
hiCP = hi95(CP)) %>%
ungroup() %>%
filter(ID == first(ID))
#plot population predictions
gp <- ggplot(data = sims2 %>% filter(row_number() != 1), aes(x=time)) +
geom_line(aes(y=medCP), col="black") +
geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
scale_y_continuous(trans = "log10") +
labs(title="Population simulation", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
PKNCA package to calculate summary statistics for
the populationlibrary(PKNCA)
# create dose object
dose_obj <-
PKNCAdose(
data,
amt~time|ID
)
# create conc object
conc_obj <-
PKNCAconc(
sims,
CP~time|ID
)
# combine
data_obj <- PKNCAdata(conc_obj, dose_obj)
## Calculate the NCA parameters
results_obj <- pk.nca(data_obj)
## Summarize the results
summary(results_obj)
## start end N auclast cmax tmax half.life aucinf.obs
## 0 24 100 26.2 [19.5] . . . .
## 0 Inf 100 . 5.33 [16.7] 1.00 [1.00, 1.00] 27.7 [7.44] 86.3 [42.4]
##
## Caption: auclast, cmax, aucinf.obs: geometric mean and geometric coefficient of variation; tmax: median and range; half.life: arithmetic mean and standard deviation; N: number of subjects
Compare doses 3 and 4 mg/kg for adults. Which dose keeps minimum voriconazole concentration at steady state above MIC (1 mg/L) for >= 90% of subjects?
library(cowplot)
data_3mg <- e %>%
as_tibble %>%
bind_cols(idata) %>%
mutate(amt = 3*BW, rate = 3*BW) %>%
select(ID, everything())
#run simulation
sims_3mg <- modA2 %>%
mrgsim_d(data=data_3mg, delta = delta, end = end, obsonly=T, outvars = c("CP")) %>%
filter(row_number() != 1)
sims2_3mg <- sims_3mg %>%
group_by(time) %>%
mutate(loCP = lo05(CP),
medCP = median(CP),
hiCP = hi95(CP)) %>%
ungroup() %>%
filter(ID == first(ID))
#plot population predictions
gp_3mg <- ggplot(data = sims2_3mg, aes(x=time)) +
geom_line(aes(y=medCP), col="black") +
geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
scale_y_continuous(trans = "log10", limits=c(0.3,7)) +
labs(title="Population simulation (3 mg/kg)", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw() +
geom_hline(aes(yintercept = 1), lty=2)
gp_4mg <- gp +
labs(title="Population simulation (4 mg/kg)", x="time (h)", y="Plasma concentration (mg/L)") +
geom_hline(aes(yintercept = 1), lty=2) +
scale_y_continuous(trans = "log10", limits=c(0.3,7))
gp_34 <- plot_grid(gp_3mg, gp_4mg, ncol=2)
gp_34
# calculate stats
stats_4mg <- sims %>%
filter(time != 0) %>%
group_by(ID) %>%
mutate(Cmin = min(CP)) %>%
slice(1) %>%
ungroup() %>%
mutate(FLG = ifelse(Cmin < 1, 1, 0))
percHi_4mg <- (nrow(stats_4mg[stats_4mg$FLG==0,]) / nrow(stats_4mg)) * 100
stats_3mg <- sims_3mg %>%
filter(time != 0) %>%
group_by(ID) %>%
mutate(Cmin = min(CP)) %>%
slice(1) %>%
ungroup() %>%
mutate(FLG = ifelse(Cmin < 1, 1, 0))
percHi_3mg <- (nrow(stats_3mg[stats_3mg$FLG==0,]) / nrow(stats_3mg)) * 100
stats_df <- tibble(Dose = c("3 mg/kg","4 mg/kg"),
`Percent of subjects with SS Cmin > MIC` = c(percHi_3mg, percHi_4mg))
stats_df %>%
knitr::kable() %>%
kable_styling()
| Dose | Percent of subjects with SS Cmin > MIC |
|---|---|
| 3 mg/kg | 71 |
| 4 mg/kg | 96 |
Use the package mrgsim.apply (vignette: https://github.com/kylebaron/mrgsim.parallel) to run the
same population simulation over parallel cores. How long does it take
compared to the single core simulation?
library(future)
library(mrgsim.parallel)
# set parallelization options
nCores <- future::availableCores()
options(future.fork.enable=TRUE, mc.cores = nCores)
plan(multicore, workers = nCores)
#run simulation
system.time(sims <- modA2 %>%
future_mrgsim_d(data=data, nchunk = nCores, delta = delta, end = end, obsonly=T, outvars = c("CP")) %>%
filter(row_number() != 1))
## user system elapsed
## 3.322 0.417 0.806
# get summary stats for population
hi95 <- function(x) quantile(x, probs = c(0.95))
lo05 <- function(x) quantile(x, probs = c(0.05))
sims2 <- sims %>%
group_by(time) %>%
mutate(loCP = lo05(CP),
medCP = median(CP),
hiCP = hi95(CP)) %>%
ungroup() %>%
filter(ID == first(ID))
#plot population predictions
gp <- ggplot(data = sims2, aes(x=time)) +
geom_line(aes(y=medCP), col="black") +
geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
scale_y_continuous(trans = "log10") +
labs(title="Population simulation - parallel", x="time (h)", y="Plasma concentration (mg/L)") +
theme_bw()
gp
Explore the simple shiny app saved as app.R
https://medcraveonline.com/MOJI/advances-in-monoclonal-antibodies-production-and-cancer-therapy.html
The story:
OPEN SCIENCE
https://ascpt.onlinelibrary.wiley.com/doi/epdf/10.1002/psp4.12461
Explore Jones model equations (https://ascpt.onlinelibrary.wiley.com/action/downloadSupplement?doi=10.1002%2Fpsp4.12461&file=psp412461-sup-0001-TableS1-S3.pdf) and model file (https://github.com/metrumresearchgroup/bioPBPK)
Results:
https://github.com/metrumresearchgroup/bioPBPK
Compile the mAb model (mAb.mod) and use it to simulate a
single 700 mg IV infusion dose of bamlanivimab infused over 2 hours.
What are the drug concentrations in plasma and lung on day 28? How do
they compare to bamlanivimab IC90 (0.41481 mcg/mL)?
mod_mab <- mread("models/mAb.mod")
# set up simulation conditions
dose <- 700/150 #700 mg to umol
dur <- 2
rate <- dose/dur
cmt <- 4
end <- 28*24
e <- ev(amt=dose, rate=rate, cmt=cmt)
sim_mab <- mod_mab %>%
mrgsim_e(e, end=end, outvars=c("Cexg_Plasma", "Cexg_Lung_IS")) %>%
as_tibble()
sim_mab2 <- sim_mab %>%
mutate(time = time/24,
Plasma = Cexg_Plasma*150,
Lung = Cexg_Lung_IS*150) %>%
select(-Cexg_Plasma, -Cexg_Lung_IS) %>%
gather(tissue, conc, -ID, -time)
# plot
p_mab <- ggplot(data=sim_mab2, aes(x=time, y=conc, col=tissue)) +
geom_line(lwd=1) +
geom_hline(yintercept = 0.41481, lty=2) +
scale_x_continuous(breaks=seq(0, 28, 7)) +
scale_y_continuous(trans = "log10") +
labs(x="Time (d)", y=expression("mAb concentration ("*mu*"g/mL)")) +
theme_bw()
p_mab
https://pubmed.ncbi.nlm.nih.gov/22143261/
Steps:
[ PARAM ] block to include tumor physiologic
parameters (volumes, flows, endothelial cell fraction, and reflection
coefficients; https://pubmed.ncbi.nlm.nih.gov/22143261/). You also
need to update PLQ_Lung to include tumor blood flow.[ MAIN ] block to include tumor FcRn in the
total FcRn calculation.[ ODE ] block to add tumor plasma and lymphatic
fluxes into the equations for plasma and lymph nodes.Build a mAb PBPK model with a simple tumor compartment following the previous steps. Run a simulation and plot mAb tumor concentration-time profile for 28 days.
lines_mod <- read_lines("models/mAb.mod")
lines_muscle <- lines_mod[str_detect(lines_mod, "Muscle")]
lines_tumor <- str_replace_all(lines_muscle, "Muscle", "Tumor")
# tumor-specific modifications
## param
lines_tumor[str_detect(lines_tumor, "^V_Tumor =")] <- "V_Tumor = 0.02 // [L] Total Volume"
lines_tumor[str_detect(lines_tumor, "^V_Tumor_IS =")] <- "V_Tumor_IS = 0.55*0.02 // [L] Interstitial Volume"
lines_tumor[str_detect(lines_tumor, "^V_Tumor_V =")] <- "V_Tumor_V = 0.07*0.02 // [L] Vascular Volume"
lines_tumor[str_detect(lines_tumor, "^PLQ_Tumor =")] <- "PLQ_Tumor = 12.7*0.02 // [L/h] blood flow rate"
lines_tumor[str_detect(lines_tumor, "^Endothelial_Cell_Frac_Tumor =")] <- "Endothelial_Cell_Frac_Tumor = 0.005"
lines_tumor[str_detect(lines_tumor, "^SV_Tumor =")] <- "SV_Tumor = 0.842"
lines_tumor[str_detect(lines_tumor, "^SIS_Tumor =")] <- "SIS_Tumor = 0.2"
## MAIN
lines_tumor[str_detect(lines_tumor, "^double FcRn_Conc =")] <- ""
## ODE
lines_tumor[str_detect(lines_tumor, "\\(1.0 - SIS_Tumor\\)")] <- ""
lines_tumor[str_detect(lines_tumor, "\\+ \\(PLQ_Tumor - LF_Tumor\\)\\*Cexg_Tumor_V")] <- "" # remove the Tumor entry to Aexg; will be patched later
# original model modifications
## PARAM
lines_mod[str_detect(lines_mod, "^PLQ_Lung = 181.913000000")] <- "PLQ_Lung = 181.9130000000000109 + (12.7*0.02) // [L/h]"
## MAIN
lines_mod[str_detect(lines_mod, "^double FcRn_Conc =")] <- paste0(str_remove(lines_mod[str_detect(lines_mod, "^double FcRn_Conc =")], "\\);"), "+V_endosomal_Tumor);")
## ODE
lines_mod[str_detect(lines_mod, "\\- L_LymphNode\\*Cedg_LN\\)/V_LN;")] <- "- L_LymphNode*Cedg_LN + (1.0 - SIS_Tumor)*LF_Tumor*Cedg_Tumor_IS)/V_LN;"
lines_mod[str_detect(lines_mod, "\\- L_LymphNode\\*Cexg_LN\\)/V_LN;")] <- "- L_LymphNode*Cexg_LN + (1.0 - SIS_Tumor)*LF_Tumor*Cexg_Tumor_IS)/V_LN;"
lines_mod[str_detect(lines_mod, "\\+ L_LymphNode\\*Cexg_LN\\);")] <- "+ L_LymphNode*Cexg_LN + (PLQ_Tumor - LF_Tumor)*Cexg_Tumor_V);"
# join lines
lines_mod_new <-
# PROB, SET, and PARAM
lines_mod[1:str_which(lines_mod, "\\[ MAIN \\]") - 1] %>%
append(lines_tumor[1:str_which(lines_tumor, "SIS_Tumor = 0.2")]) %>%
# MAIN
append(lines_mod[str_which(lines_mod, "\\[ MAIN \\]"):(str_which(lines_mod, "\\[ CMT \\]") - 1)]) %>%
append(lines_tumor[str_which(lines_tumor, "double LF_Tumor = PLQ_Tumor\\*0.002"):str_which(lines_tumor, "CFcRn_Tumor_IM_0 = FcRn_Conc\\*1e-4;")]) %>%
# CMT
append(lines_mod[str_which(lines_mod, "\\[ CMT \\]"):(str_which(lines_mod, "\\[ ODE \\]") - 1)]) %>%
append(lines_tumor[str_which(lines_tumor, "^Cedg_Tumor_V$"):str_which(lines_tumor, "^CFcRn_Tumor_IM$")]) %>%
# ODE
append(lines_mod[str_which(lines_mod, "\\[ ODE \\]"):(str_which(lines_mod, "\\[ CAPTURE \\]") - 1)]) %>%
append(lines_tumor[str_which(lines_tumor, "dxdt_Cexg_Tumor_V ="):str_which(lines_tumor, "2.0\\*kdeg_FcRn_Ab\\*Cedg_Tumor_b2IM\\)\\)/V_Tumor_IM;")]) %>%
# CAPTURE
append(lines_mod[str_which(lines_mod, "\\[ CAPTURE \\]"):length(lines_mod)])
# save
write_lines(lines_mod_new, "models/mAb_tumor.mod")
# compile
mod_mab_tumor <- mread("models/mAb_tumor.mod")
# run simulation
# set up simulation conditions
dose <- 700/150 #700 mg to umol
dur <- 2
rate <- dose/dur
cmt <- 4
end <- 28*24
e <- ev(amt=dose, rate=rate, cmt=cmt)
sim_mab_tumor <- mod_mab_tumor %>%
mrgsim_e(e, end=end, outvars=c("Cexg_Plasma", "Cexg_Tumor_IS")) %>%
as_tibble()
sim_mab_tumor2 <- sim_mab_tumor %>%
mutate(time = time/24,
Plasma = Cexg_Plasma*150,
Tumor = Cexg_Tumor_IS*150) %>%
select(-Cexg_Plasma, -Cexg_Tumor_IS) %>%
gather(tissue, conc, -ID, -time)
# plot
p_mab_tumor <- ggplot(data=sim_mab_tumor2 %>% filter(tissue != "Lung"), aes(x=time, y=conc, col=tissue)) +
geom_line(lwd=1) +
scale_x_continuous(breaks=seq(0, 28, 7)) +
scale_y_continuous(trans = "log10") +
labs(x="Time (d)", y=expression("mAb concentration ("*mu*"g/mL)")) +
theme_bw()
p_mab_tumor
PBPKToolkit is an open-source R package that provides a set of functions to be used for PBPK modeling. The functions mainly generate drug- and system-specific parameters required to build a PBPK model. The package is still in early stages of development so submitting any issues would be greatly appreciated to help improve it. This package does not belong to any organization.
library(PBPKToolkit)
Two functions genInd and genPop can be used
to generate a virtual individual or a population, respectively, for PBPK
modeling of small molecules. The algorithms implemented are adapted from
Willmann et al 2007 source and Huisinga
et al 2012 source.
These methods maintains a correlation between the physiologic parameters
and the sampled individual’s covariates to generate realistic
individuals. The algorithm is guided by these databases:
# define the individual demographics
## You only need to define two of the three covariates: body weight, height, and BMI
age <- 30
ismale <- TRUE
bw <- 73
ht <- 1.76
# generate individual physiological parameters
## Willmann
indPars <- genInd(age=age, is.male=ismale, bw_targ=bw, ht_targ=ht, optimize = FALSE, method="Willmann")
## Huisinga
indPars <- genInd(age=age, is.male=ismale, bw_targ=bw, ht_targ=ht, method="Huisinga")
## run simulation
modA %>% ev(e) %>% param(indPars) %>% mrgsim() %>% plot
The function calcKp can calculate the molecule’s Kp
values for different organs using one of five different calculation
methods:
The function uses the standardized tissue composition database reported here
# define molecule's physicochemical properties
logP <- 2 #lipophilicity
pKa <- 1 #acidic strength
type <- 3 #type of molecule
BP <- 1 #blood:plasma concentration ratio
fup <- 0.5 #unbound fraction in plasma
method <- "PT" #prediction method
# calculate partition coefficients
Kps <- calcKp(logP=logP, pKa=pKa, fup=fup, BP=BP, type=type, method=method)
If Vss is available, it can be passed to the argument
Vss of the calcKp function that will then
scale the calculated Kp values accordingly. If Vss is
provided, the named list containing the individual’s physiological
parameters also need to be passed to the argument Vt.
calcKp will grab the tissue volumes from this list to allow
for the scaling of the Kp values based on the Vss calculation here https://jcheminf.biomedcentral.com/articles/10.1186/s13321-015-0054-x.
This scaling also requires Kpot, which is the partition
coefficient value for the “other” compartment that lumps all
compartments that are not defined in the standardized tissue composition
database. If Kpot is left as NULL, it will be
calculated as the average of all non adipose tissues Kp values.
# calculate partition coefficients
Vss <- 50
indPars <- genInd(age=age, is.male=ismale, bw_targ=bw, ht_targ=ht, optimize = FALSE)
Kps <- calcKp(logP=logP, pKa=pKa, fup=fup, BP=BP, type=type, method=method, Vss=Vss, Vt=indPars)
In case BP parameter was missing, the function calcBP
can be used to calculate BP based on the methods reported here. There are two
methods to chose from:
Note: drug type = “total” is the default type and it uses the regression coefficients calculated by fitting different molecule types (acids, bases, and neutrals) together.
# in case BP parameter was missing, the function calcBP
calcBP(fup = fup, method = 1)
## [1] 0.827023
calcBP(logP = logP, fup = fup, method = 2)
## [1] 1.034007
In case fup parameter was missing, the function calcFup
can be used to calculate fup using the molecule’s logP (or another
measurement of lipophilicity like logD) based on the method reported here.
calcFup(logP = logP)
## [1] 0.2931778
Predict Vss from estimated partition coefficients and tissue volumes.
calcVss(Kp=Kps, BP=BP, Vt=indPars)
## [1] 50